(i) Ø and any variable is a modal sentence.
(ii) If A is a modal sentence, then []A is a modal sentence.
(iii) If A is a modal sentence and B is a modal sentence, then A implies B (A→B) is a modal sentence.

Because Ø indicates contradiction, Ø is always false. But by the truth table of material implication, A → Ø is true if and only if either A is false or Ø is true. But Ø can’t be true. So A → Ø is true if and only if A is false.

This symbol ∞ will stand for something being proved.

(1) Hypothesis (HY) : A new hypothesis may be added to a proof anytime, but the hypothesis begins a new sub-proof.

(2) Modus Ponens (MP) : If A implies B and A, then B must lie in exactly the same sub-proof.

(3) Conditional Proof (CP): When proof of B is derived from the hypothesis A, it follows that A implies B, where A implies B lies outside hypothesis A.

(4) Double Negation (DN): Removal of double negation ~~A & A lie in the same same sub-proof.

(5) Reiteration (R): Sentence A may be copied into a new sub-proof.

Proof of Modus Tollens: Given the conditional claim that the consequent is true if the antecedent is true, and given that the consequent is false, we can infer that the antecedent is also false.